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Shelah's categoricity conjecture from a successor for tame abstract elementary classes

Published online by Cambridge University Press:  12 March 2014

Rami Grossberg  and
Monica Vandieren
Rami Grossberg
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh
Monica Vandieren
Affiliation:
Department of Mathematics, University of Michigan
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Abstract

We prove a categoricity transfer theorem for tame abstract elementary classes.

Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K+}. If K is categorical in λ and λ+, then K is categorical in λ++.

Combining this theorem with some results from [37]. we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes:

Suppose K is χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ0 ≔ Hanf(K). Ifand K is categorical in somethen K is categorical in μ for all μ .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 2 , June 2006 , pp. 553 - 568
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Baldwin, John, Abstract elementary classes, Monograph, in preparation. Available at http://www2.math.uic.edu/~jbaldwin/model.html.Google Scholar
[2]Baldwin, John, Non-splitting extensions. Technical report, available at http://www2.math.uic.edu/~jbaldwin/model.html.Google Scholar
[3]Baldwin, John, Kueker, David, and VanDieren, Monica, Upward stability transfer for tame abstract elementary classes. Submitted, http://www.math.lsa.umich.edu/~mvd/home.html.Google Scholar
[4]Ben-Yaacov, Itay, Positive model theory and compact abstract theories, Journal of Mathematical Logic, vol. 3 (2003), no. 1, pp. 85–118.CrossRefGoogle Scholar
[5]Berenstein, Alexander, Some generalizations of first order tools to homogeneous models, this Journal. To appear. Preprint available at http://www.math.uiuc.edu/~aberenst/research.html.Google Scholar
[6]Berenstein, Alexander and Buechler, Steven, A study of independence in strongly homogeneous expansions of Hilbert spaces. Preprint available at http://www.math.uiuc.edu/~aberenst/research.html.Google Scholar
[7]Buechler, Steven and Lessmann, Olivier, Simple homogeneous models, Journal of the American Mathematical Society, vol. 16 (2003), no. 1. pp. 91–121.CrossRefGoogle Scholar
[8]Grossberg, Rami, Classification theory for non-elementary classes, Logic and algebra (Zhang, Yi, editor), Contemporary Mathematics, vol. 302, American Mathematical Society, 2002, pp. 165–204.CrossRefGoogle Scholar
[9]Grossberg, Rami and Lessmann, Olivier, Shelah's stability spectrum and homogeneity spectrum in finite diagrams. Archives in Mathematical Logic, vol. 41 (2002), no. 1. pp. 1–31.CrossRefGoogle Scholar
[10]Grossberg, Rami and VanDieren, Monica, Categoricity from one successor cardinal in tame abstract elementary classes, 17 pages. Submitted, http://www.math.lsa.umich.edu/~mvd/home.html.Google Scholar
[11]Grossberg, Rami and VanDieren, Monica, Galois-stability in tame abstract elementary classes. 23 pages. Submitted in 10/4/2004. http://www.math.cmu.edu/~rami.Google Scholar
[12]Hart, Bradd and Shelah, Saharon, Categoricity over P for first order T or categoricity for ϕ ϵ Lω1ω can stop at ℵk while holding for ℵ0, …, ℵk−1. Israel Journal of Mathematics, vol. 70 (1990). pp. 219–235.Google Scholar
[13]Henson, C. Ward and Iovino, José, Ultraproducts in analysis. Analysis and Logic, London Mathematical Society Lecture Note Series, Cambridge University Press, to appear. Part I of the three part book by Henson, C. W., Iovino, J., Kechris, A. S., and Odell, E. W..CrossRefGoogle Scholar
[14]Hyttinen, Tapani, Generalizing Morley's theorem, Mathematical Logic Quarterly, vol. 44 (1998), pp. 176–184.CrossRefGoogle Scholar
[15]Hyttinen, Tapani, On nonstructure of elementary submodels of a stable homogeneous structure, Fundamenta Mathematical vol. 156 (1998), pp. 167–182.CrossRefGoogle Scholar
[16]Hyttinen, Tapani and Shelah, Saharon, Strong splitting in stable homogeneous models, Annals of Pure and Applied Logic, vol. 103 (2000), pp. 201–228.CrossRefGoogle Scholar
[17]Iovino, José, Stable Banach spaces and Banach space structures, I: Fundamentals, Models, algebras, and proofs (Caicedo, X. and Montenegro, C., editors). Marcel Dekker, New York, 1999, pp. 97–113.Google Scholar
[18]Iovino, José, Stable Banach spaces and Banach space structures, II: Forking and compact topologies, Models, algebras, and proofs (Caicedo, X. and Montenegro, C., editors). Marcel Dekker, New York, 1999. pp. 77–95.Google Scholar
[19]Jónsson, Bjarni, Homogeneous universal systems, Mathematica Scandinavica, vol. 8 (1960), pp. 137–142.CrossRefGoogle Scholar
[20]Keisler, H. Jerome, Lω1ω(Q). Annals of Mathematical Logic, vol. 1 (1969).Google Scholar
[21]Keisler, H. Jerome, Model theory for infinitary logic, North-Holland, 1971.Google Scholar
[22]Kolman, Oren and Shelah, Saharon, Categoricity of Theories in Lk,ω when κ is a measurable cardinal, Part I. Fundamentae Mathematicae, vol. 151 (1996), pp. 209–240.Google Scholar
[23]Lessmann, Olivier, Pregeometries infinite diagrams, Annals of Pure and Applied Logic, vol. 106, (2000). no. 1–3. pp. 49–83.CrossRefGoogle Scholar
[24]Lessmann, Olivier, Upward categoricity from a successor cardinal for tame abstract classes with amalgamation, this Journal, vol. 70 (2005). no. 2, pp. 639–660.Google Scholar
[25]Łoś, Jerzy, On the categoricity in power of elementary deductive systems and related problems, Colloquium Mathematicum, vol. 3 (1954), pp. 58–62.CrossRefGoogle Scholar
[26]Makkai, Michael and Shelah, Saharon, Categoricity of theories Lkω with κ a compact cardinal, Annals of Pure and Applied Logic, vol. 47 (1990), pp. 41–97.Google Scholar
[27]Marcus, Leo, A prime minimal model with an infinite set of indiscernibles, Israel Journal of Mathematics, vol. 11 (1972), pp. 180–183.CrossRefGoogle Scholar
[28]Morley, Michael, Categoricity in power, Transactions of the American Mathematical Society. vol. 114 (1965). pp. 514–538.CrossRefGoogle Scholar
[29]Shelah, Saharon, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69–118.CrossRefGoogle Scholar
[30]Shelah, Saharon, Categoricity of uncountable theories, Proceedings of the Tarski Symposium (Henkin, L. A.et al., editors), AMS, Providence, R.I., 1974, pp. 187–203.CrossRefGoogle Scholar
[31]Shelah, Saharon, Categoricity in ℵ1 of sentences in (Q). Israel Journal of Mathematics, (1975). pp. 127–148.CrossRefGoogle Scholar
[32]Shelah, SaharonThe lazy model-theoretician's guide to stability, Logique et Analyse, vol. 18 (1975), pp. 241–308.Google Scholar
[33]Shelah, Saharon, Classification theory for nonelementary classes. I. The number of uncountable models of Ψ ϵ Lω1ω, Part A, Israel Journal of Mathematics, vol. 46 (1983). pp. 212–240.CrossRefGoogle Scholar
[34]Shelah, Saharon, Classification theory for nonelementary classes. I. The number of uncountable models of Ψ ϵ Lω1ω. Part B. Israel Journal of Mathematics, vol. 46 (1983). pp. 241–273.CrossRefGoogle Scholar
[35]Shelah, Saharon, Classification of nonelementary classes. II. Abstract elementary classes, Classification theory, Lecture Notes in Mathematics, vol. 1292. Springer-Berlin, 1987, pp. 419–497.CrossRefGoogle Scholar
[36]Shelah, Saharon, Classification theory and the number of non-isomorphic models, 2 ed., North Holland, Amsterdam, 1990.Google Scholar
[37]Shelah, Saharon, Categoricity of abstract classes with amalgamation, Annals of Pure and Applied Logic, vol. 98 (1999). no. 1–3, pp. 241–294.CrossRefGoogle Scholar
[38]Shelah, Saharon, On what I do not understand (and have something to say), model theory, Mathematica Japonica, vol. 51 (2000), pp. 329–377.Google Scholar
[39]Shelah, Saharon, Categoricity of an abstract elementary class in two successive cardinals, Israel Journal of Mathematics, vol. 126 (2001), pp. 29–128.CrossRefGoogle Scholar
[40]Shelah, Saharon, Categoricity of theories in Lκ*ω when κ* is a measurable cardinal. Part II. Dedicated to the memory of Jerzy Łoś, Fundamenta Mathematica, vol. 170 (2001). no. 1–2, pp. 165–196.CrossRefGoogle Scholar
[41]Shelah, Saharon, Categoricity in abstract elementary classes: going up inductive step. Preprint. 100 pages.Google Scholar
[42]Shelah, Saharon, Toward classification theory of good λ frames and abstract elementary classes.Google Scholar
[43]Shelah, Saharon and Villaveces, Andrés. Categoricity in abstract elementary classes with no maximal models, Annals of Pure and Applied Logic, vol. 97 (1999). no. 1–3. pp. 1–25.CrossRefGoogle Scholar
[44]VanDieren, Monica, Categoricity in abstract elementary classes with no maximal models, Annals of Pure and Applied Logic. 61 pages, accepted, http://www.math.Isa.umich.edu/~mvd/home.html.Google Scholar
[45]Zilber, Boris, Analytic and pseudo-analytic structures, Preprint, http://www.maths.ox.ac.uk/~zilber.Google Scholar